D40:1C4 - GroupNames

G = D₄₀⋊₁C₄ order 320 = 2⁶·5

1^st semidirect product of D₄₀ and C₄ acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₄₀⋊₁C₄, Q₁₆⋊₂F₅, D₁₀.₃D₈, Dic₅.₁₇SD₁₆, C₈.₃(C₂×F₅), C₄₀.₃(C₂×C₄), C₄₀⋊C₄⋊₂C₂, (C₅×Q₁₆)⋊₁C₄, C₅⋊₂(D₈⋊₂C₄), (C₄×D₅).₂₄D₄, C₅⋊₂C₈.₁₅D₄, C₈.F₅⋊₃C₂, C₄.₅(C₂²⋊F₅), Q₈.D₁₀.₂C₂, C₂₀.₅(C₂²⋊C₄), (C₈×D₅).₁₁C₂², C₁₀.₉(D₄⋊C₄), C₂.₁₀(D₂₀⋊C₄), SmallGroup(320,245)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₄₀ — D₄₀⋊₁C₄

Chief series

C₁ — C₅ — C₁₀ — C₂₀ — C₄×D₅ — C₈×D₅ — C₄₀⋊C₄ — D₄₀⋊₁C₄

Lower central

C₅ — C₁₀ — C₂₀ — C₄₀ — D₄₀⋊₁C₄

Upper central

C₁ — C₂ — C₄ — C₈ — Q₁₆

Generators and relations for D₄₀⋊₁C₄
G = < a,b,c | a⁴⁰=b²=c⁴=1, bab=a^-1, cac^-1=a³, cbc^-1=a³⁷b >

Subgroups: 346 in 58 conjugacy classes, 20 normal (all characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₅, C₈, C₈, C₂×C₄, D₄, Q₈, D₅, C₁₀, C₁₆, C₄⋊C₄, C₂×C₈, D₈, SD₁₆, Q₁₆, C₄○D₄, Dic₅, C₂₀, C₂₀, F₅, D₁₀, D₁₀, C₄.Q₈, M₅(2), C₄○D₈, C₅⋊₂C₈, C₄₀, C₄×D₅, C₄×D₅, D₂₀, C₅×Q₈, C₂×F₅, D₈⋊₂C₄, C₅⋊C₁₆, C₈×D₅, D₄₀, Q₈⋊D₅, C₅×Q₁₆, C₄⋊F₅, Q₈⋊₂D₅, C₈.F₅, C₄₀⋊C₄, Q₈.D₁₀, D₄₀⋊₁C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂²⋊C₄, D₈, SD₁₆, F₅, D₄⋊C₄, C₂×F₅, D₈⋊₂C₄, C₂²⋊F₅, D₂₀⋊C₄, D₄₀⋊₁C₄

Character table of D₄₀⋊₁C₄

class	1	2A	2B	2C	4A	4B	4C	4D	4E	5	8A	8B	8C	10	16A	16B	16C	16D	20A	20B	20C	40A	40B
size	1	1	10	40	2	8	10	40	40	4	4	10	10	4	20	20	20	20	8	16	16	8	8
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	1	-1	-1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	1	linear of order 2
ρ₃	1	1	1	-1	1	-1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	-1	-1	1	1	linear of order 2
ρ₄	1	1	1	-1	1	-1	1	-1	-1	1	1	1	1	1	1	1	1	1	1	-1	-1	1	1	linear of order 2
ρ₅	1	1	-1	1	1	-1	-1	i	-i	1	1	-1	-1	1	-i	i	i	-i	1	-1	-1	1	1	linear of order 4
ρ₆	1	1	-1	-1	1	1	-1	i	-i	1	1	-1	-1	1	i	-i	-i	i	1	1	1	1	1	linear of order 4
ρ₇	1	1	-1	1	1	-1	-1	-i	i	1	1	-1	-1	1	i	-i	-i	i	1	-1	-1	1	1	linear of order 4
ρ₈	1	1	-1	-1	1	1	-1	-i	i	1	1	-1	-1	1	-i	i	i	-i	1	1	1	1	1	linear of order 4
ρ₉	2	2	2	0	2	0	2	0	0	2	-2	-2	-2	2	0	0	0	0	2	0	0	-2	-2	orthogonal lifted from D₄
ρ₁₀	2	2	-2	0	2	0	-2	0	0	2	-2	2	2	2	0	0	0	0	2	0	0	-2	-2	orthogonal lifted from D₄
ρ₁₁	2	2	2	0	-2	0	-2	0	0	2	0	0	0	2	√2	-√2	√2	-√2	-2	0	0	0	0	orthogonal lifted from D₈
ρ₁₂	2	2	2	0	-2	0	-2	0	0	2	0	0	0	2	-√2	√2	-√2	√2	-2	0	0	0	0	orthogonal lifted from D₈
ρ₁₃	2	2	-2	0	-2	0	2	0	0	2	0	0	0	2	-√-2	-√-2	√-2	√-2	-2	0	0	0	0	complex lifted from SD₁₆
ρ₁₄	2	2	-2	0	-2	0	2	0	0	2	0	0	0	2	√-2	√-2	-√-2	-√-2	-2	0	0	0	0	complex lifted from SD₁₆
ρ₁₅	4	4	0	0	4	-4	0	0	0	-1	4	0	0	-1	0	0	0	0	-1	1	1	-1	-1	orthogonal lifted from C₂×F₅
ρ₁₆	4	4	0	0	4	4	0	0	0	-1	4	0	0	-1	0	0	0	0	-1	-1	-1	-1	-1	orthogonal lifted from F₅
ρ₁₇	4	4	0	0	4	0	0	0	0	-1	-4	0	0	-1	0	0	0	0	-1	-√5	√5	1	1	orthogonal lifted from C₂²⋊F₅
ρ₁₈	4	4	0	0	4	0	0	0	0	-1	-4	0	0	-1	0	0	0	0	-1	√5	-√5	1	1	orthogonal lifted from C₂²⋊F₅
ρ₁₉	4	-4	0	0	0	0	0	0	0	4	0	-2√-2	2√-2	-4	0	0	0	0	0	0	0	0	0	complex lifted from D₈⋊₂C₄
ρ₂₀	4	-4	0	0	0	0	0	0	0	4	0	2√-2	-2√-2	-4	0	0	0	0	0	0	0	0	0	complex lifted from D₈⋊₂C₄
ρ₂₁	8	8	0	0	-8	0	0	0	0	-2	0	0	0	-2	0	0	0	0	2	0	0	0	0	orthogonal lifted from D₂₀⋊C₄, Schur index 2
ρ₂₂	8	-8	0	0	0	0	0	0	0	-2	0	0	0	2	0	0	0	0	0	0	0	-√10	√10	orthogonal faithful, Schur index 2
ρ₂₃	8	-8	0	0	0	0	0	0	0	-2	0	0	0	2	0	0	0	0	0	0	0	√10	-√10	orthogonal faithful, Schur index 2

Smallest permutation representation of D₄₀⋊₁C₄
►On 80 points

Generators in S₈₀

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)

(1 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 72)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 65)(17 64)(18 63)(19 62)(20 61)(21 60)(22 59)(23 58)(24 57)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)

(2 28 10 4)(3 15 19 7)(5 29 37 13)(6 16)(8 30 24 22)(9 17 33 25)(11 31)(12 18 20 34)(14 32 38 40)(23 35 39 27)(26 36)(41 66 61 46)(42 53 70 49)(43 80 79 52)(44 67 48 55)(45 54 57 58)(47 68 75 64)(50 69 62 73)(51 56 71 76)(59 72 63 60)(65 74 77 78)

G:=sub<Sym(80)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,80)(2,79)(3,78)(4,77)(5,76)(6,75)(7,74)(8,73)(9,72)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41), (2,28,10,4)(3,15,19,7)(5,29,37,13)(6,16)(8,30,24,22)(9,17,33,25)(11,31)(12,18,20,34)(14,32,38,40)(23,35,39,27)(26,36)(41,66,61,46)(42,53,70,49)(43,80,79,52)(44,67,48,55)(45,54,57,58)(47,68,75,64)(50,69,62,73)(51,56,71,76)(59,72,63,60)(65,74,77,78)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,80)(2,79)(3,78)(4,77)(5,76)(6,75)(7,74)(8,73)(9,72)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,63)(19,62)(20,61)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41), (2,28,10,4)(3,15,19,7)(5,29,37,13)(6,16)(8,30,24,22)(9,17,33,25)(11,31)(12,18,20,34)(14,32,38,40)(23,35,39,27)(26,36)(41,66,61,46)(42,53,70,49)(43,80,79,52)(44,67,48,55)(45,54,57,58)(47,68,75,64)(50,69,62,73)(51,56,71,76)(59,72,63,60)(65,74,77,78) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,80),(2,79),(3,78),(4,77),(5,76),(6,75),(7,74),(8,73),(9,72),(10,71),(11,70),(12,69),(13,68),(14,67),(15,66),(16,65),(17,64),(18,63),(19,62),(20,61),(21,60),(22,59),(23,58),(24,57),(25,56),(26,55),(27,54),(28,53),(29,52),(30,51),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41)], [(2,28,10,4),(3,15,19,7),(5,29,37,13),(6,16),(8,30,24,22),(9,17,33,25),(11,31),(12,18,20,34),(14,32,38,40),(23,35,39,27),(26,36),(41,66,61,46),(42,53,70,49),(43,80,79,52),(44,67,48,55),(45,54,57,58),(47,68,75,64),(50,69,62,73),(51,56,71,76),(59,72,63,60),(65,74,77,78)]])

Matrix representation of D₄₀⋊₁C₄ ►in GL₈(𝔽₂₄₁)

0	0	1	240	0	0	0	0
0	0	1	0	0	0	0	0
240	0	1	0	0	0	0	0
0	240	1	0	0	0	0	0
0	0	0	0	222	19	0	0
0	0	0	0	222	222	0	0
0	0	0	0	192	114	38	95
0	0	0	0	13	129	137	0

234	124	117	7	0	0	0	0
117	0	124	7	0	0	0	0
234	7	124	0	0	0	0	0
0	7	117	124	0	0	0	0
0	0	0	0	192	114	38	95
0	0	0	0	192	114	0	95
0	0	0	0	222	19	0	0
0	0	0	0	135	59	116	176

0	0	240	0	0	0	0	0
240	0	0	0	0	0	0	0
0	0	0	240	0	0	0	0
0	240	0	0	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	240	0	0
0	0	0	0	49	127	0	146
0	0	0	0	206	194	137	0

G:=sub<GL(8,GF(241))| [0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,1,1,1,1,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,0,222,222,192,13,0,0,0,0,19,222,114,129,0,0,0,0,0,0,38,137,0,0,0,0,0,0,95,0],[234,117,234,0,0,0,0,0,124,0,7,7,0,0,0,0,117,124,124,117,0,0,0,0,7,7,0,124,0,0,0,0,0,0,0,0,192,192,222,135,0,0,0,0,114,114,19,59,0,0,0,0,38,0,0,116,0,0,0,0,95,95,0,176],[0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,1,0,49,206,0,0,0,0,0,240,127,194,0,0,0,0,0,0,0,137,0,0,0,0,0,0,146,0] >;

D₄₀⋊₁C₄ in GAP, Magma, Sage, TeX

D_{40}\rtimes_1C_4

% in TeX

G:=Group("D40:1C4");

// GroupNames label

G:=SmallGroup(320,245);

// by ID

G=gap.SmallGroup(320,245);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,387,675,794,80,1684,851,102,6278,3156]);

// Polycyclic

G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^37*b>;

// generators/relations

Export

Character table of D₄₀⋊₁C₄ in TeX

0	0	1	240	0	0	0	0
0	0	1	0	0	0	0	0
240	0	1	0	0	0	0	0
0	240	1	0	0	0	0	0
0	0	0	0	222	19	0	0
0	0	0	0	222	222	0	0
0	0	0	0	192	114	38	95
0	0	0	0	13	129	137	0

234	124	117	7	0	0	0	0
117	0	124	7	0	0	0	0
234	7	124	0	0	0	0	0
0	7	117	124	0	0	0	0
0	0	0	0	192	114	38	95
0	0	0	0	192	114	0	95
0	0	0	0	222	19	0	0
0	0	0	0	135	59	116	176

0	0	240	0	0	0	0	0
240	0	0	0	0	0	0	0
0	0	0	240	0	0	0	0
0	240	0	0	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	240	0	0
0	0	0	0	49	127	0	146
0	0	0	0	206	194	137	0

0	0	1	240	0	0	0	0
0	0	1	0	0	0	0	0
240	0	1	0	0	0	0	0
0	240	1	0	0	0	0	0
0	0	0	0	222	19	0	0
0	0	0	0	222	222	0	0
0	0	0	0	192	114	38	95
0	0	0	0	13	129	137	0

234	124	117	7	0	0	0	0
117	0	124	7	0	0	0	0
234	7	124	0	0	0	0	0
0	7	117	124	0	0	0	0
0	0	0	0	192	114	38	95
0	0	0	0	192	114	0	95
0	0	0	0	222	19	0	0
0	0	0	0	135	59	116	176

0	0	240	0	0	0	0	0
240	0	0	0	0	0	0	0
0	0	0	240	0	0	0	0
0	240	0	0	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	240	0	0
0	0	0	0	49	127	0	146
0	0	0	0	206	194	137	0

G = D40⋊1C4 order 320 = 26·5

1st semidirect product of D40 and C4 acting faithfully

G = D₄₀⋊₁C₄ order 320 = 2⁶·5

1^st semidirect product of D₄₀ and C₄ acting faithfully

0	0	1	240	0	0	0	0
0	0	1	0	0	0	0	0
240	0	1	0	0	0	0	0
0	240	1	0	0	0	0	0
0	0	0	0	222	19	0	0
0	0	0	0	222	222	0	0
0	0	0	0	192	114	38	95
0	0	0	0	13	129	137	0

234	124	117	7	0	0	0	0
117	0	124	7	0	0	0	0
234	7	124	0	0	0	0	0
0	7	117	124	0	0	0	0
0	0	0	0	192	114	38	95
0	0	0	0	192	114	0	95
0	0	0	0	222	19	0	0
0	0	0	0	135	59	116	176

0	0	240	0	0	0	0	0
240	0	0	0	0	0	0	0
0	0	0	240	0	0	0	0
0	240	0	0	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	240	0	0
0	0	0	0	49	127	0	146
0	0	0	0	206	194	137	0